3.884 \(\int (2+e x)^{3/2} \sqrt{12-3 e^2 x^2} \, dx\)

Optimal. Leaf size=65 \[ -\frac{2 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{16 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{32 (2-e x)^{3/2}}{\sqrt{3} e} \]

[Out]

(-32*(2 - e*x)^(3/2))/(Sqrt[3]*e) + (16*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) - (2*Sqrt
[3]*(2 - e*x)^(7/2))/(7*e)

_______________________________________________________________________________________

Rubi [A]  time = 0.0827166, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{16 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{32 (2-e x)^{3/2}}{\sqrt{3} e} \]

Antiderivative was successfully verified.

[In]  Int[(2 + e*x)^(3/2)*Sqrt[12 - 3*e^2*x^2],x]

[Out]

(-32*(2 - e*x)^(3/2))/(Sqrt[3]*e) + (16*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) - (2*Sqrt
[3]*(2 - e*x)^(7/2))/(7*e)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.9798, size = 51, normalized size = 0.78 \[ - \frac{32 \left (- 3 e x + 6\right )^{\frac{3}{2}}}{9 e} - \frac{2 \sqrt{3} \left (- e x + 2\right )^{\frac{7}{2}}}{7 e} + \frac{16 \sqrt{3} \left (- e x + 2\right )^{\frac{5}{2}}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+2)**(3/2)*(-3*e**2*x**2+12)**(1/2),x)

[Out]

-32*(-3*e*x + 6)**(3/2)/(9*e) - 2*sqrt(3)*(-e*x + 2)**(7/2)/(7*e) + 16*sqrt(3)*(
-e*x + 2)**(5/2)/(5*e)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0458328, size = 50, normalized size = 0.77 \[ \frac{2 (e x-2) \sqrt{4-e^2 x^2} \left (15 e^2 x^2+108 e x+284\right )}{35 e \sqrt{3 e x+6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + e*x)^(3/2)*Sqrt[12 - 3*e^2*x^2],x]

[Out]

(2*(-2 + e*x)*Sqrt[4 - e^2*x^2]*(284 + 108*e*x + 15*e^2*x^2))/(35*e*Sqrt[6 + 3*e
*x])

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 44, normalized size = 0.7 \[{\frac{ \left ( 2\,ex-4 \right ) \left ( 15\,{e}^{2}{x}^{2}+108\,ex+284 \right ) }{105\,e}\sqrt{-3\,{e}^{2}{x}^{2}+12}{\frac{1}{\sqrt{ex+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+2)^(3/2)*(-3*e^2*x^2+12)^(1/2),x)

[Out]

2/105*(e*x-2)*(15*e^2*x^2+108*e*x+284)*(-3*e^2*x^2+12)^(1/2)/(e*x+2)^(1/2)/e

_______________________________________________________________________________________

Maxima [A]  time = 0.797866, size = 81, normalized size = 1.25 \[ \frac{{\left (30 i \, \sqrt{3} e^{3} x^{3} + 156 i \, \sqrt{3} e^{2} x^{2} + 136 i \, \sqrt{3} e x - 1136 i \, \sqrt{3}\right )}{\left (e x + 2\right )} \sqrt{e x - 2}}{105 \,{\left (e^{2} x + 2 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(3/2),x, algorithm="maxima")

[Out]

1/105*(30*I*sqrt(3)*e^3*x^3 + 156*I*sqrt(3)*e^2*x^2 + 136*I*sqrt(3)*e*x - 1136*I
*sqrt(3))*(e*x + 2)*sqrt(e*x - 2)/(e^2*x + 2*e)

_______________________________________________________________________________________

Fricas [A]  time = 0.21868, size = 84, normalized size = 1.29 \[ -\frac{2 \,{\left (15 \, e^{5} x^{5} + 78 \, e^{4} x^{4} + 8 \, e^{3} x^{3} - 880 \, e^{2} x^{2} - 272 \, e x + 2272\right )}}{35 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(3/2),x, algorithm="fricas")

[Out]

-2/35*(15*e^5*x^5 + 78*e^4*x^4 + 8*e^3*x^3 - 880*e^2*x^2 - 272*e*x + 2272)/(sqrt
(-3*e^2*x^2 + 12)*sqrt(e*x + 2)*e)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+2)**(3/2)*(-3*e**2*x**2+12)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-3 \, e^{2} x^{2} + 12}{\left (e x + 2\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(3/2), x)